Congratulations you're officially about to become a ski racer Mom/Dad
Everything you need for race registration can be found at the PARA website.
You'll need a race entry form, USSA membership card, your checkbook some envelopes and stamps. If your child will attend the State Championships he/she must be a PARA member as well.
Each race has slightly different requirements for how they accept and process race registrations. They all use the same race entry form for registration. Some require 1 check per racer per form per day, some don't allow faxed entries, many have different date cutoffs, many don't allow phone entries, etc. Each race lists its requirements on the PARA race schedule on the PARA website along with the address the entry cards and payments should be mailed to. You'll be able to see what age groups run during a race, what the entry fees are, where to send them and any other notes they have added re the registration process specific to that race/mountain/race club. You should note the schedule and cutoff dates for races. On site registration is almost always impossible.
You can view the schedule by going to the PARA website and clicking on Race Info and then Race Schedules. At the PARA website you can also see updates for dates/race rescheduling, postponements, cancellations, etc. The PARA website is a very important resource for you.
That's pretty much it. If you have additional questions ask Dan, Steve or Ian.